{"id":"62a7cd51-fafc-4b5e-85c9-27be47f6bd7d","arxiv_id":"2509.05736","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A Koopman-operator-based monitor shrinks the RED step size whenever a low-dimensional model of the iterates predicts divergence, preventing PSNR collapse.","lead":"The authors add a stability monitor to RED, an image reconstruction algorithm that repeatedly applies a pretrained denoiser. The monitor models the iterate dynamics with a low-dimensional linear operator and shrinks the step size when the model warns of divergence, keeping reconstructions stable over long runs.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Controller's decision rule assumes ρ(K_t) from a 66-d OLS feature fit tracks RED's true stability; without a bound or direct validation, this is the weakest load-bearing link.","rationale":"Reader's weakest_assumption correctly identifies the absence of a link between ρ(K_t) and full-system stability. My reading supports this: Sec. 2.3 asserts the indicator without proof, Sec. 2.4 describes the feature map by sensitivity studies, and the experiments do not measure whether ρ(K_t) tracks the true unstable modes. I considered other possible objections—small test set, no error bars, hyperparameter tuning—but they are secondary to the decision rule. If the Jacobian comparison test shows high agreement, the method's empirical support is much stronger. If it shows disagreement, the central mechanism is not validated even though the method might still work sometimes. I recommend no change to the reader's CONDITIONAL verdict; the concern is real but addressable, exactly as the verdict states.","tokens_in":9545,"tokens_out":4529,"duration_ms":55749,"concrete_test":"Run 20 unstable RED trajectories (e.g., motion deblurring with DnCNN/DRUNet, 8 Levin kernels, 256×256). At every checkpoint t∈Ω, compute ρ(K_t) via Eq. (7) and independently estimate the spectral radius of the true Jacobian DT_{γ_t}(x_t) using power iteration with Jacobian-vector products (autodiff through the denoiser). Record agreement between ρ(K_t)>1 and true spectral radius >1 (ROC/AUC or simple confusion counts). Repeat with a 6-dimensional feature map (global mean/std only). If agreement is high and the reduced map fails, the premise is supported; if ρ(K_t) disagrees with true expansion in a substantial fraction of checkpoints, the controller is acting on a feature-space artifact rather than the instability it claims to detect.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Algorithm 1 makes a single load-bearing decision: shrink γ iff ρ(K_t)>1. K_t is obtained by OLS (Eq. 7) in a 66-dimensional handcrafted feature space, and ρ(K_t) is used as if it were a proxy for the local expansion rate of the full RED map T_γ (Eq. 4). The paper provides no theorem, bound, or error estimate connecting ρ(K_t) to the spectral properties of DT_γ. If the feature map's span is not approximately invariant under the Koopman operator, the OLS operator can miss genuinely unstable modes (false negative: no slowdown, instability persists) or introduce spurious expanding modes (false positive: unnecessary step-size decay and slow convergence). The empirical evidence is suggestive but not conclusive: 15 test images, no error bars, and the feature map/window/stride/β are tuned on the same tasks. Because step-size control is the whole mechanism, this gap is the central risk to the claim that SKOOP 'achieves consistent stability improvements.'","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes SKOOP-RED, an adaptive step-size controller for Regularization-by-Denoising (RED) image reconstruction. At checkpoints during the RED iteration, the method projects recent iterates into a 66-dimensional handcrafted feature space, fits a low-dimensional Koopman operator K_t by least squares (Eq. 7), and shrinks the RED step size whenever the spectral radius ρ(K_t) exceeds 1 (Algorithm 1). The authors claim this stabilizes RED with black-box deep denoisers, avoids retraining, adds less than 20% runtime overhead, and outperforms Vanilla RED and Equivariant RED across deblurring and superresolution tasks with DnCNN, DRUNet, GS-DRUNet, and DiffUNet. Code is provided.","tokens_in":9919,"tokens_out":4457,"duration_ms":53964,"significance":"If the empirical claims hold, this is a practically useful contribution: RED with modern deep denoisers is known to be unstable, and a model-agnostic, low-overhead stabilization mechanism would strengthen the case for RED in real imaging systems. The paper is clearly written, the experiments cover four denoisers and three tasks, and the code release is a concrete asset. However, the central mechanism—using the spectral radius of a 66-dimensional OLS-fitted Koopman operator as a proxy for the stability of the full RED dynamics—is asserted without a supporting bound or direct validation. The evaluation also tunes key hyperparameters on the same test tasks and reports no variance over the 15-image set. These gaps are load-bearing for the claim of 'consistent stability improvements,' but they are addressable with additional analysis and a more rigorous evaluation protocol.","major_comments":[{"comment":"The controller's decision rule assumes that ρ(K_t) ≥ 1 indicates potential instability of the full RED map T_γ in R^n. This link is not derived. K_t is an OLS fit in a low-dimensional feature space; without an error bound relating the approximate spectrum to the spectral properties of DT_γ, the feature map can miss genuinely unstable modes (false negatives, no step-size reduction) or introduce spurious expanding modes (false positives, unnecessary slowdown). The paper's own appendix (Section A) describes the feature map as 'empirically' useful, which effectively concedes that this link is heuristic. To make the central claim load-bearing, the authors should provide either a formal error estimate in terms of the feature-map invariance/closure and the residual of Eq. (7), or direct validation on small-scale problems comparing ρ(K_t) against the spectral radius of the full Jacobian along th","section":"Sec. 2.3, Eqs. (7)-(8)"},{"comment":"The stability claim rests on 15 test images (Set15C), and Tables 2 and S1 report only mean PSNR values with no variance, percentiles, or per-image breakdowns. Equally important, the feature map, window size w, stride r, and decay rate β are selected via sensitivity studies on the same tasks (Section 2.4, Fig. S2), creating a risk of indirect overfitting to the evaluation set. I request: (i) report standard deviations or per-image results; (ii) freeze all SKOOP parameters on a development split and evaluate on held-out tasks, kernel types, or denoisers; (iii) include an ablation that replaces ρ(K_t) with a simple heuristic (e.g., residual-norm growth or a fixed decay schedule) to demonstrate that the Koopman spectral-radius monitoring, rather than mere step-size adaptation, is what provides the stability benefit.","section":"Sec. 3, Tables 2 and S1; Sec. 2.4"},{"comment":"The control law only shrinks γ and never increases it. This is not necessarily wrong, but it means any false positive permanently slows the iteration, and the paper does not discuss recovery from overly conservative step sizes. Moreover, the resulting non-autonomous iteration is not analyzed at all; the only theoretical framing is the heuristic spectral-radius test. A formal statement for a simplified or linearized model (e.g., under what conditions the adaptive scheme keeps γ bounded away from zero and prevents divergence) would substantially strengthen the paper and clarify what SKOOP-RED can guarantee.","section":"Algorithm 1 and Eq. (8)"}],"minor_comments":[{"comment":"Equation (6) states K_t ψ(x_τ) = ψ(x_{τ+1}) for t−w+1 ≤ τ ≤ t, but for τ = t this would require the future iterate x_{t+1}. Equation (7) correctly sums only up to τ = t−1. Please fix the index range in Eq. (6) to avoid confusion.","section":"Sec. 2.3, Eq. (6) vs. Eq. (7)"},{"comment":"The text says SKOOP-RED adds 'only a small 15–20% runtime overhead per iteration,' but Table 1 reports overheads of 10.97–14.11%, and Section 2.4 reports 11% for DiffUNet. Please reconcile these numbers.","section":"Sec. 3, Table 1"},{"comment":"The abbreviation 'Equiv.RED' is used without a definition in the table caption; it appears to refer to equivariant RED from [18], but this should be stated explicitly.","section":"Table 2"},{"comment":"The paper says K_t is computed using the technique in [33] (DMD), but Eq. (7) is a least-squares fit. Please specify which DMD variant is used (e.g., exact DMD, projected DMD) and what rank/regularization, if any, is applied.","section":"Sec. 2.4"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the scope of a letters journal and the empirical results are promising, but the missing theoretical link between the low-dimensional spectral radius and full RED stability is the core risk. I would like to see either a concrete bound or a direct Jacobian-based validation, plus a more stringent evaluation protocol that separates parameter selection from testing. These are substantial but feasible revisions; the current version is not yet ready for acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper does something new and useful — it fits an online Koopman/DMD model to RED iterates in a 66-dimensional feature space and shrinks the step size when the learned spectral radius exceeds 1. Across four pretrained denoisers and several inverse problems, it consistently prevents the divergence that vanilla RED shows, with code, sensitivity studies, and measured overhead under 20%. I'd send this to a referee.\n\nWhat's actually new: the combination is new, even though the ingredients are not. DMD and the Koopman operator are standard, and RED's instability is well documented. Using a low-dimensional spectral-radius estimate as an adaptive step-size trigger for iterative reconstruction is a legitimate first, and the paper does not oversell it as a theory of RED convergence.\n\nWhat it does well: the experiments are reasonably broad — motion deblurring, Gaussian deblurring, superresolution, four different denoiser architectures — and the comparison to equivariant RED is fair. The runtime breakdown is honest, the code is public, and the sensitivity analysis for window size, stride, and beta gives the reader a sense of how fragile the method is. The supplementary long-run experiment (10k iterations) is a nice touch.\n\nSoft spots, in order of importance. First, the load-bearing link between the spectral radius of the OLS-fitted 66-dimensional operator and the actual stability of the full RED map is asserted, not derived. The stress-test note is right that a feature map can miss an unstable mode or create a spurious one. But I don't think this is fatal: the paper is an empirical letter, not a convergence analysis, and the ablation shows the trigger does something. Second, hyperparameters (feature map, window, stride, beta) are tuned on the same tasks used in the main tables, and the PSNR numbers are averages over 15 images with no error bars. That limits the strength of the claim, but again it is addressable in revision. Third, there is no discussion of failure cases — instances where the trigger fired but divergence still happened, or where vanilla RED was stable and the controller slowed things down unnecessarily. That would genuinely strengthen the paper.\n\nI disagree with any suggestion that this is circular or that the citation pattern is problematic. It is an adaptive controller, not a derivation of the result from the target. Self-citations are appropriate and not excessive.\n\nThe bottom line: this is a pragmatic, clearly described stabilization mechanism for a real practical problem. It belongs in the iterative-imaging literature, and a careful referee could improve it substantially. My recommendation: accept for peer review, and ask the authors for error bars, one or two failure-case analyses, and ideally a small experiment connecting the learned spectral radius to the full Jacobian's spectral behavior. For people using RED with black-box denoisers, this is a useful safety net and worth reading.","headline":"A solid empirical letter: the Koopman/DMD step-size controller is genuinely new for RED, the experiments are consistent, and the main weakness is that the stability proxy is heuristic rather than proven.","tokens_in":10285,"tokens_out":2057,"would_cite":true,"duration_ms":27207,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Adaptive Koopman monitor stops RED reconstruction from diverging","keywords":["Koopman operator","Regularization by Denoising","adaptive step size","image reconstruction","spectral radius","deep denoisers","stability","dynamic mode decomposition"],"falsifier":"Construct or find a pretrained denoiser and task where, at the moment vanilla RED begins to diverge, rho(K_t) remains below 1, so SKOOP-RED never shrinks gamma; if PSNR then drops while the monitor reports stability, the spectral-radius criterion is falsified. Conversely, if rho(K_t) exceeds 1 during runs that remain stable, the controller is over-conservative.","tokens_in":9502,"feed_emoji":"🖼️","tokens_out":3973,"duration_ms":36394,"temperature":0.7,"pith_summary":"The paper claims that RED (Regularization-by-Denoising), a popular scheme that uses pretrained denoisers as regularizers for image reconstruction, can be stabilized without retraining by monitoring a low-dimensional Koopman operator fitted to its recent iterates. When the spectral radius of that operator exceeds 1, the step size is shrunk exponentially. Across four deep denoisers and three imaging tasks, this prevents the PSNR divergence that vanilla RED exhibits and keeps reconstructions artifact-free. The appeal is that the fix is model-agnostic: it treats the denoiser as a black box and only adjusts the iteration step size, at less than 20 percent added runtime.","feed_headline":"Koopman operator steers RED to stable reconstructions","feed_subtitle":"A 66-dimensional spectral-radius check shrinks the step size before deep denoisers diverge, with under 20 percent overhead.","key_machinery":"The Koopman operator K_t, approximated in a 66-dimensional observable space by ordinary least squares from a window of recent RED iterates. The paper uses the spectral radius of K_t as a proxy for the local contraction rate of the full RED iteration in R^n; values above 1 trigger an exponential step-size reduction. The feature map (global statistics, 4x4 pooling, low-frequency DCT) is the handcrafted observable choice that makes the low-dimensional fit informative.","core_discovery":"SKOOP-RED is an adaptive step-size schedule for RED. At checkpoints, the most recent iterates are projected into a 66-dimensional feature space combining per-channel mean and standard deviation, 4x4 block means, and low-frequency DCT coefficients. A linear operator K_t is fitted by least squares to advance these features one step, approximating the Koopman operator of the RED dynamics; its spectral radius rho(K_t) is read as a local stability indicator. If rho(K_t) >= 1, the step size gamma is multiplied by exp(-beta(rho-1)) with beta = 2; otherwise gamma is unchanged. The paper reports that this simple rule produces stable, high-PSNR reconstructions where vanilla RED diverges, across DnCNN,","pith_inferences":["The stability indicator could be sharpened by comparing rho(K_t) against the spectral radius of the full Jacobian on a few examples; if they disagree, a richer observable set would be needed.","A more principled choice of the feature map and window, rather than sensitivity-based selection, might yield a bound connecting the finite-dimensional Koopman spectrum to the true RED dynamics.","The same checkpointed Koopman monitor may generalize to other iterative reconstruction algorithms, such as plug-and-play ADMM or diffusion-based restoration, where divergence also appears after many iterations.","Since the controller only shrinks the step size, it cannot fix a denoiser that is unstable in every direction; the method's ceiling is the quality of the underlying denoiser."],"forward_implications":["RED can be run with arbitrary pretrained deep denoisers without imposing nonexpansiveness or retraining.","The adaptive step size removes the need for early stopping at an unpredictable breakdown point.","The same denoiser and forward model can be used across tasks; only the Koopman monitor needs to be fitted online.","The overhead is modest enough for iterative restoration in practice.","The approach is not tied to gradient descent structure and could be transferred to other RED variants."],"supporting_citations":[{"why":"Introduces RED and its fixed-step iterations, the algorithm being stabilized.","marker":"[13]"},{"why":"Shows deep denoisers can violate the assumptions underlying RED and cause instability, motivating the stabilization.","marker":"[14]"},{"why":"Introduces the Koopman operator, the theoretical basis for lifting nonlinear dynamics to a linear operator.","marker":"[27]"},{"why":"Connects Koopman spectral analysis to finite-dimensional data-driven approximations used here.","marker":"[29]"},{"why":"Supplies dynamic mode decomposition, the least-squares procedure used to fit K_t.","marker":"[33]"},{"why":"Equivariant RED is the main baseline against which SKOOP-RED is compared.","marker":"[18]"},{"why":"Provides DRUNet, one of the pretrained denoisers used in the experiments.","marker":"[15]"},{"why":"Provides DnCNN, another pretrained denoiser used in the experiments.","marker":"[36]"},{"why":"Provides DiffUNet, used as a diffusion-based denoiser in the experiments.","marker":"[37]"}],"fun_headline_variants":["Koopman spectral radius keeps RED from diverging","Adaptive step-size rule tames RED instability","Koopman operator stabilizes RED without retraining","Spectral-radius check prevents RED blow-up","Data-driven stability for RED via Koopman"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The result rests on the assumption that the spectral radius of a 66-dimensional least-squares linear operator fitted to recent RED iterates in a handcrafted feature space is a faithful early warning of whether the full RED iteration in image space is about to diverge.","fun_headline_variants_meta":{"raw":{"variants":["Koopman spectral radius keeps RED from diverging","Adaptive step-size rule tames RED instability","Koopman operator stabilizes RED without retraining","Spectral-radius check prevents RED blow-up","Data-driven stability for RED via Koopman"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000139,"raw_usage":{"total_tokens":961,"prompt_tokens":675,"completion_tokens":286,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":419,"completion_tokens_details":{"reasoning_tokens":214}},"tokens_in":419,"tokens_out":286,"duration_ms":3578,"temperature":1.0,"reasoning_tokens":214,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T05:05:09.637610+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct or find a pretrained denoiser and task where, at the moment vanilla RED begins to diverge, rho(K_t) remains below 1, so SKOOP-RED never shrinks gamma; if PSNR then drops while the monitor reports stability, the spectral-radius criterion is falsified. Conversely, if rho(K_t) exceeds 1 during runs that remain stable, the controller is over-conservative.","supporting_citations":[{"cited_title":"The little engine that could: Regularization by Denoising (RED).SIAM J","cited_arxiv_id":null,"evidence_quote":"Introduces RED and its fixed-step iterations, the algorithm being stabilized."},{"cited_title":"Regularization by denoising: Clarifications and new interpretations.IEEE T rans","cited_arxiv_id":null,"evidence_quote":"Shows deep denoisers can violate the assumptions underlying RED and cause instability, motivating the stabilization."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Koopman operator, the theoretical basis for lifting nonlinear dynamics to a linear operator."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Connects Koopman spectral analysis to finite-dimensional data-driven approximations used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies dynamic mode decomposition, the least-squares procedure used to fit K_t."},{"cited_title":"Terris, T","cited_arxiv_id":null,"evidence_quote":"Equivariant RED is the main baseline against which SKOOP-RED is compared."},{"cited_title":"Zhang, Y","cited_arxiv_id":null,"evidence_quote":"Provides DRUNet, one of the pretrained denoisers used in the experiments."},{"cited_title":"Zhang, W","cited_arxiv_id":null,"evidence_quote":"Provides DnCNN, another pretrained denoiser used in the experiments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides DiffUNet, used as a diffusion-based denoiser in the experiments."}],"review_version":1}